Circumconic and inconic
Suppose A,B,C are distinct non-collinear points, and let ΔABC denote the triangle whose vertices are A,B,C. Following common practice, A denotes not only the vertex but also the angle BAC at vertex A, and similarly for B and C as angles in ΔABC. Let a = |BC|, b = |CA|, c = |AB|, the sidelengths of ΔABC.
In trilinear coordinates, the general circumconic is the locus of a variable point X = x : y : z satisfying an equation
- uyz + vzx + wxy = 0,
for some point u : v : w. The isogonal conjugate of each point X on the circumconic, other than A,B,C, is a point on the line
- ux + vy + wz = 0.
This line meets the circumcircle of ΔABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
The general inconic is tangent to the three sidelines of ΔABC and is given by the equation
- u2x2 + v2y2 + w2z2 − 2vwyz − 2wuzx − 2uvxy = 0.
Centers and tangent lines
The center of the general circumconic is the point
- u(−au + bv + cw) : v(au − bv + cw) : w(au + bv − cw).
The lines tangent to the general circumconic at the vertices A,B,C are, respectively,
- wv + vz = 0,
- uz + wx = 0,
- vx + uy = 0.
The center of the general inconic is the point
- cy + bz : az + cx : bx + ay.
The lines tangent to the general inconic are the sidelines of ΔABC, given by the equations x = 0, y = 0, z = 0.
- Each noncircular circumconic meets the circumcircle of ΔABC in a point other than A, B, and C, often called the fourth point of intersection, given by trilinear coordinates
- (cx − az)(ay − bx) : (ay − bx)(bz − cy) : (bz − cy)(cx − az)
- If P = p : q : r is a point on the general circumconic, then the line tangent to the conic at P is given by
- (vr + wq)x + (wp + ur)y + (uq + vp)z = 0.
- The general circumconic reduces to a parabola if and only if
- u2a2 + v2b2 + w2c2 − 2vwbc − 2wuca − 2uvab = 0,
and to a rectangular hyperbola if and only if
- x cos A + y cos B + z cos C = 0.
- The general inconic reduces to a parabola if and only if
- ubc + vca + wab = 0.
- Suppose that p1 : q1 : r1 and p2 : q2 : r2 are distinct points, and let
- X = (p1 + p2t) : (q1 + q2t) : (r1 + r2t).
As the parameter t ranges through the real numbers, the locus of X is a line. Define
- X2 = (p1 + p2t)2 : (q1 + q2t)2 : (r1 + r2t)2.
The locus of X2 is the inconic, necessarily an ellipse, given by the equation
- L4x2 + M4y2 + N4z2 − 2M2N2yz − 2N2L2zx − 2L2M2xy = 0,
- L = q1r2 − r1q2,
- M = r1p2 − p1r2,
- N = p1q2 − q1p2.
- A point in the interior of a triangle is the center of an inellipse of the triangle if and only if the point lies in the interior of the triangle whose vertices lie at the midpoints of the original triangle's sides.:p.139
- The lines connecting the tangency points of the inellipse of a triangle with the opposite vertices of the triangle are concurrent.:p.148
- Of all triangles inscribed in a given ellipse, the centroid of the one with greatest area coincides with the center of the ellipse.:p.147
Extension to quadrilaterals
All the centers of inellipses of a given quadrilateral fall on the line segment connecting the midpoints of the diagonals of the quadrilateral.:p.136
- Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.